Title: Obstructions to Lifting Tropical Curves in Surfaces in 3-Space
Abstract:Tropicalization is a procedure that takes subvarieties of an algebraic torus to balanced weighted rational complexes in space. In this paper, we study the tropicalizations of curves in surfaces in $3$...Tropicalization is a procedure that takes subvarieties of an algebraic torus to balanced weighted rational complexes in space. In this paper, we study the tropicalizations of curves in surfaces in $3$-space. These are balanced rational weighted graphs in tropical surfaces. Specifically, we study the lifting problem: given a graph in a tropical surface, can one find a corresponding algebraic curve in a surface? We develop specific combinatorial obstructions to lifting a graph by reducing the problem to the question of whether one can factor a polynomial with particular support in the characteristic $0$ case. This explains why some unusual tropical curves constructed by Vigeland are not liftable.Read More